Crystalline materials, which have periodic structure, can be used to reflect x-rays based on diffraction. The reflection of x-rays from crystal planes can only occur when the Bragg condition is met: EQU 2d sin .theta.=n.lambda.
Where .lambda. is the x-ray wavelength, d is the spacing of reflection planes, .theta. is the incident angle with respect to the reflection planes, and n is the reflection order. The d spacings for natural crystals and most synthetic crystals are constant. In order to reflect x-rays of the same wavelength efficiently, a crystal optical element must have a near constant incident angle with respect to the reflection planes of the crystal on every point of the surface. Crystal optics based on Bragg reflection have been widely used for x-ray monochromators and high-resolution spectroscopy. However, the applications of crystal optics for focusing and collimating x-rays from a laboratory source have been limited because of the strict requirement of the Bragg condition and the narrow rocking curve widths for most useful crystalline materials.
For many applications of microanalysis, an intense monochromatic x-ray beam based on a laboratory type source is needed. Three-dimensional focusing of x-rays from a laboratory source involves doubly bent crystal optics. The practical use of a toroidal crystal to focus 8 ke V x-rays has been demonstrated recently with the use of a mica crystal based on the Johann type point to point focusing geometry. For example, reference an article by Z. W. Chen and D. B. Wittry entitled "Microanalysis by Monochromatic Microprobe X-ray Fluorescence-Physical Basis, Properties and Future Prospects", J. Appl. Phys., 84(2), page 1064 (1998). However, the Bragg condition cannot be satisfied on every point of the crystal using this approach due to the geometrical aberration of the Johann geometry in the Roland circle plane, which will limit the collection solid angle of the optic. The spot size of the focused beam is also limited by the geometrical aberration of the toroidal surface.
On the other hand, a parallel monochromatic x-ray beam is useful for many x-ray diffraction applications. Conventional crystal optics with constant d spacing cannot provide efficient collimation of hard x-rays from a divergent source since the incident angle must vary from point to point for any type of collimating mirror. For high-resolution x-ray diffraction applications, the monochromaticity provided by conventional multilayer optics is relatively poor and the divergence is not small enough.